A triangle has corners at #(1 ,9 )#, #(-6 ,-1 )#, and #(1 ,-2 )#. If the triangle is dilated by a factor of #2 # about point #(-5 ,-2 ), how far will its centroid move?

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Answer 1 · 2018-05-25T10:02:36

#color(blue)((sqrt(265))/3" units")#

If all vertices of a triangle are dilated by a factor #bba# then the centroid will also be dilated by a factor #bba#

Let #A# be the triangle in its original position.

Let #A'# be the dilated triangle.

Centroid of #A#

The coordinates of the centroid can be found by taking the arithmetic mean of the x coordinates and the y coordinates.

#((x_1+x_2+x_3)/3,(y_1+y_2+y_3)/3)#

#((1-6+1)/3,(9-1-2)/3)=>(-4/3,2)#

Using vectors:

Position vector of centre of dilation:

#vec(OD)=((-5),(-2))#

Let #C# be the centroid of #A'#.

Vector from D to C:

#vec(DC)=((11/3),(4))#

Scale factor = 2:

Position vector of new centroid #C'#

#vec(OD)+3vec(DC)=((-5),(-2))+2((11/3),(4))=((7/3),(6))#

Distance the centroid has moved can be found using the distance formula:

#d=sqrt((-4/3-7/3)^2+(2-6)^2)=(sqrt(265))/3# units